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Knot Symmetries and the Fundamental Quandle

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Knot Symmetries and the Fundamental Quandle KNOT SYMMETRIES AND THE FUNDAMENTAL QUANDLE EVA HORVAT Abstract. We establish a relationship between the knot symmetries and the automorphisms of the knot quandle. We identify the homeomorphisms of the pair (S3;K) that induce the (anti)automorphisms of the fundamental quandle Q(K). We show that every quandle (anti)automorphism of Q(K) is induced by a homeomorphism of the pair (S3;K). As an application of those results, we are able to explore some symmetry properties of a knot based on the presentation of its fundamental quandle, which is easily derived from a knot diagram. 1. Introduction It is well known that the knot quandle is a complete knot invariant [8]. The knot quandle and various derived quandle invariants have been extensively used to study and distinguish nonequivalent knots. Our idea is to study knot equivalences from the viewpoint of the funda- mental quandle. In this paper, we establish a relationship between the knot symmetries and the automorphisms of the knot quandle. Our goal is to investigate the complex topological information, hidden in the group of knot symmetries, from a purely algebraical perspective of the knot quandle. Our main results are the following. 3 3 Proposition 1.1. Let f :(S ;K) ! (S ;K) be a homeomorphism for which [fj@NK ] = ±1 2 2 MCG(T ). Then f induces a map f∗ : Q(K) ! Q(K) that is either a quandle automorphism or a quandle antiautomorphism. Proposition 1.2. Let F : Q(K) ! Q(K) be an (anti)automorphism of the fundamental quan- dle of a nontrivial knot K. Then F is induced by a homeomorphism that preserves the orienta- tion of the normal bundle if F is an automorphism, and reverses the orientation of the normal bundle if F is an antiautomorphism. arXiv:1707.04824v1 [math.GT] 16 Jul 2017 This paper is organized as follows. In the Section 2, we introduce the basic concepts from the theory of knot quandles that will be needed in the rest of the paper. Subsection 2.1 contains the definition of the quandle, the augmented quandle, the associated group and some basics about the quandle homomorphisms. In the Subsection 2.2 we define the quandle presentations. In the Subsection 2.3, we define the fundamental quandle of a knot in S3, note some of its properties and describe its presentation. The Section 3 contains the core results of the paper. Recalling the basics of knot symmetries, we first establish how and when a knot symmetry induces a knot quandle (anti)automorphism. Then we use a Theorem of Matveev to show that Date: January 3, 2018. 2010 Mathematics Subject Classification. 57M27, 57M05 (primary), 57M25 (secondary). Key words and phrases. knots, knot symmetries, fundamental quandle. 1 each knot quandle (anti)automorphism is induced by a knot symmetry. In the final Section 4, we do some calculations displaying the use of our results in investigating the symmetries of knots. We provide a Phyton code for calculating the Q-group of symmetries from an alternating planar diagram of a knot. 2. Preliminaries 2.1. The definition of a quandle. Definition 2.1. [4] A quandle is an algebraic structure comprising a nonempty set Q with two binary operations (x; y) 7! x B y and (x; y) 7! x C y, which satisfies three axioms: (1) x B x = x. (2)( x B y) C y = x = (x C y) B y. (3)( x B y) B z = (x B z) B (y B z). It follows from the second axiom that the map Sy : Q ! Q defined by Sy(x) = x B y is a bijection for any y 2 Q, and by the third axiom it is an automorphism of Q. By the first quandle axiom the automorphism Sy fixes y. Sy is called an inner automorphism of Q. The subgroup Inn(Q) of all the inner automorphisms is a normal subgroup of Aut(Q) [4]. −1 Example 2.2. Let G be a group. Define the operations B; C: G × G ! G by a B b = b ab −1 and a C b = bab for any a; b 2 G. It is easy to see that these operations satisfy all the quandle axioms, and the resulting quandle is denoted by Gconj. Thus we obtain a quandle from any group by forgetting multiplication and setting both conjugations as the operations. Example 2.3. For any quandle Q = (S; B; C), the triple (S; C; B) defines another quandle, which we denote by Qd and is called the dual quandle of Q. Definition 2.4. Let G be a group, acting on itself by conjugation as gh := h−1gh for any g; h 2 G. An augmented quandle (X; G) is a set X with an action by the group G, written as (x; g) 7! xg and a function @ : X ! G, satisfying the augmentation conditions (1) x@x = x for all x 2 X, (2) @(xg) = g−1(@x)g for all x 2 X; g 2 G. @y (@y)−1 The quandle operations on X are then defined by x B y := x and x C y := x . Definition 2.5. Let Q1 and Q2 be quandles. A quandle homomorphism from Q1 to Q2 is a map f : Q1 ! Q2 satisfying f(x B y) = f(x) B f(y) for any x; y 2 Q1. A quandle antihomomorphism from Q1 to Q2 is a map g : Q1 ! Q2 satisfying g(xBy) = g(x) C g(y) for any x; y 2 Q1. Remark 2.6. Observe that a quandle antihomomorphism g : Q1 ! Q2 is actually a quandle d homomorphism from Q1 to Q2. Lemma 2.7. If f : Q1 ! Q2 is a quandle homomorphism, then f(x C y) = f(x) C f(y). If f : Q1 ! Q2 is a quandle antihomomorphism, then f(x C y) = f(x) B f(y). 2 Proof. Let f : Q1 ! Q2 be a quandle homomorphism and choose x; y 2 Q1. Denoting w = xCy, we compute w B y = x ) f(w) B f(y) = f(x) ) f(x C y) = f(w) = f(x) C f(y). A similar proof settles the case when f is a quandle antihomomorphism. Lemma 2.8. For any quandle Q, the set AutC (Q) = ff : Q ! Qj f a bijection; f a homomorphism or an antihomomorphismg with the composition operation forms a group. Proof. The set Sym(Q) of all bijective maps of the set Q to itself forms a group for compo- sition. We would like to show that AutC (Q) ⊂ Sym(Q) is a subgroup. Thus, we need to show that AutC (Q) is closed under composition and invertation. A composition of two quan- C dle automorphisms is obviously a quandle automorphism. Now let g1; g2 2 Aut (Q) be two antihomomorphisms, and choose any x; y 2 Q. We may compute: x B y = z ) g2(x) C g2(y) = g2(z) ) g2(z) B g2(y) = g2(x) ) g1(g2(z)) C g1(g2(y)) = g1(g2(x)) ) g1(g2(x)) B g1(g2(y)) = g1(g2(z)) ; thus g1 ◦ g2 is a quandle automorphism. In a similar way we show that a composition of a quandle homomorphism and a quandle antihomomorphism (or vice versa) is a quandle antiho- momorphism. The inverse of a quandle automorphism is a quandle automorphism. The inverse of any quandle antihomomorphism g is a quandle antihomomorphism, since their composition is the identity automorphism: x B y = z ) g(x) C g(y) = g(z) ) g(z) B g(y) = g(x) ) −1 −1 −1 −1 g (g(z) B g(y)) = x = z C y ) g (g(z) B g(y)) = g (g(z)) C g (g(y)) : Definition 2.9. Let Q be a quandle. The associated group As(Q) of the quandle Q is defined as As(Q) = F (Q)=K, where F (Q) is the free group, generated by the set Q, and K is the normal −1 −1 −1 −1 subgroup of F (Q), given by K = f(a B b)b a b; (a C b)ba b j a; b 2 Qg. Lemma 2.10. Let Q be a quandle and let η : Q ! As(Q) be the natural map. Let G be a group with the conjugation quandle Gconj and the natural map @ : Gconj ! G. Given any quandle homomorphism f : Q ! Gconj, there exists a unique group homomorphism f# : As(Q) ! G such that f# ◦ η = @ ◦ f. η Q As(Q) f f# @ Gconj G Proof. Let As(Q) = F (Q)=K, where K is the normal subgroup from the Definition 2.9. The quandle homomorphism f induces a map φ: F (Q) ! G. By the definition of a quandle −1 homomorphism, the quandle operations of the Example 2.2 yield φ(aBb) = φ(b) φ(a)φ(b) and 3 −1 φ(a C b) = φ(b)φ(a)φ(b) for every a; b 2 F (Q), thus the normal subgroup K lies in the kernel of φ. It follows that φ factors through a unique group homomorphism f# : As(Q) ! G. 2.2. Quandle presentations. We recall the following from [7]. Let A be a set we call the alphabet, whose elements we call the letters.A word in A is any finite sequence, consisting of letters and the symbols (; ); B and C. Define inductively the set D(A) of admissible words according to the rules: (1) For any a 2 A, the word a is admissible, (2) If w and z are admissible words, then w B z and w C z are also admissible, (3) There are no other admissible words except those obtained inductively by the rules (1) and (2).
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